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convergence condition of infinite product

Let us think the sequenceu1,u1⁢u2,u1⁢u2⁢u3,…subscriptu1subscriptu1subscriptu2subscriptu1subscriptu2subscriptu3normal-…u_{1},\,u_{1}u_{2},\,u_{1}u_{2}u_{3},\,\ldots In the complex analysis, one often uses the definition of the convergence of an infinite product∏k=1∞uksuperscriptsubscriptproductk1subscriptuk\displaystyle\prod_{{k=1}}^{{\infty}}u_{k} where the case limk→∞⁡u1⁢u2⁢…⁢uk=0subscriptnormal-→ksubscriptu1subscriptu2normal-…subscriptuk0\displaystyle\lim_{{k\to\infty}}u_{1}u_{2}\ldots u_{k}=0 is excluded. Then one has the

Theorem.

The infinite product∏k=1∞uksuperscriptsubscriptproductk1subscriptuk\displaystyle\prod_{{k=1}}^{{\infty}}u_{k} of the non-zerocomplex numbersu1subscriptu1u_{1}, u2subscriptu2u_{2}, … is convergentiff for every positivenumberεε\varepsilon there exists a positive number nεsubscriptnεn_{\varepsilon} such that the condition

When the infinite product converges, we say that the value of the infinite product is equal to limk→∞⁡u1⁢u2⁢…⁢uksubscriptnormal-→ksubscriptu1subscriptu2normal-…subscriptuk\displaystyle\lim_{{k\to\infty}}u_{1}u_{2}\ldots u_{k}.